There are 7 Greek island groups of which 3, the Saronic, Sporades, and Cyclades are closest to Athens and most suitable for those short on time. Also close by is beautiful Euboea, the 2nd largest Greek island and only an hour from Athens by train, bus or car.
Extremely time constrained visitors should consider the 1 day 3 island Saronic Gulf Cruise or the regularly scheduled Saronic islands ferry service. The Saronic Islands are the closest Greek island to Athens (you could actually swim to Saronic Salamis if you wanted to but its not recommended for several reasons). The closeness of several of the Saronic islands may be ascertained personally from the heights of the Acropolis or Mt. Lykavittos the two best views in town.
From Athens, the Ionian and Dodecanese groups are many nautical miles distant and require more of a time commitment or a flight. There is no one best way to enter Greece. The best way for you will depend on many factors. If you are flexible then consider the most common ways to enter Greece by rail, boat, flight or bus.
In the central Aegean, the Cyclades group is perhaps the most famous and consists of 55 islands, 24 of which are inhabited, of which Kea, is closest to Athens (3 hrs via Lavrion). The Cyclades is far enough offshore to give a true sense of the Greek islands and offers, in summer, good inter-island connections between its major islands. What's more, most of its islands are relatively small and impart a true "I am on an island feeling" unlike Crete, Euboea and Rhodes.
The majority of Aegean islands are served by Athens' main port, known today as Piraeus and known in antiquity as Kantharos. Besides its main harbor, Piraeus has two smaller harbors, Zea and Micro-limani (Munychia). From Zea high speed hydrofoils depart for Saronic Gulf and Peloponnesian destinations. Athens secondary ports of Lavrion and Rafina also help to alleviate the congestion of Piraeus and offer certain advantages to travelers depending on destination but tend to operate, as do all ferries and flights, most frequently in summer.
For an at a glance overview of Greek Islands ferry service and schedules click here.
See the 18 or so Greek Islands with airports.
There are also islands so near the mainland that driving is possible: Euboea (or Evia, 2hrs) due north of Athens, Poros & Spetses (4 & 5 hrs) southwest and Lefkada northwest (6hrs).
Similarly, to reach the Sporades from Athens one must travel by land to the ports of Aghios Konstandinos, Kimi or Volos and then board ship.
Crete which has several ports, may be reached via Piraeus or by air. In summer Crete is also accessible via ferry from Gythion on the Peloponnese.
From Crete itself one may reach both the Cyclades and the Dodecanese with varying frequency by air and sea. Crete has several ports with Heraklion offering the most frequent connections.